Space- and Time-Dependent Quantum Dynamics of Spatially Indirect Excitons in Semiconductor Heterostructures Federico Grasselli, Andrea Bertoni, and Guido Goldoni

Home , Potential well

Space- and time-dependent quantum dynamics of spatially indirect excitons in semiconductor heterostructures Federico Grasselli, Andrea Bertoni, and Guido Goldoni

Citation: The Journal of Chemical Physics 142, 034701 (2015); doi: 10.1063/1.4905483 View online: http://dx.doi.org/10.1063/1.4905483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/3?ver=pdfcov Published by the AIP Publishing

Articles you may be interested in Dynamics of indirect excitons in a coupled quantum-well pair J. Appl. Phys. 113, 153706 (2013); 10.1063/1.4801808

On Bose condensation of excitons in quasi-two-dimensional semiconductor heterostructures Low Temp. Phys. 38, 541 (2012); 10.1063/1.4733681

Tunneling escape time from a semiconductor quantum well in an electric field J. Appl. Phys. 106, 113701 (2009); 10.1063/1.3259414

Ambipolar diffusion and spatial and time-resolved spectroscopies in semiconductor heterostructures J. Appl. Phys. 106, 043503 (2009); 10.1063/1.3173176

Temperature dependence of exciton transfer in hybrid quantum well/nanocrystal heterostructures Appl. Phys. Lett. 91, 092126 (2007); 10.1063/1.2776865

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 THE JOURNAL OF CHEMICAL PHYSICS 142, 034701 (2015)

Space- and time-dependent quantum dynamics of spatially indirect excitons in semiconductor heterostructures Federico Grasselli,1,2,a) Andrea Bertoni,2,b) and Guido Goldoni1,2,c) 1Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Modena, Italy 2CNR-NANO S3, Institute for Nanoscience, Via Campi 213/a, 41125 Modena, Italy (Received 10 November 2014; accepted 22 December 2014; published online 15 January 2015)

We study the unitary propagation of a two-particle one-dimensional Schrödinger equation by means of the Split-Step Fourier method, to study the coherent evolution of a spatially indirect exciton (IX) in semiconductor heterostructures. The mutual Coulomb interaction of the electron-hole pair and the electrostatic potentials generated by external gates and acting on the two particles separately are taken into account exactly in the two-particle dynamics. As relevant examples, step/downhill and barrier/well potential profiles are considered. The space- and time-dependent evolutions during the scattering event as well as the asymptotic time behavior are analyzed. For typical parameters of GaAs- based devices, the transmission or reflection of the pair turns out to be a complex two-particle process, due to comparable and competing Coulomb, electrostatic, and kinetic energy scales. Depending on the intensity and anisotropy of the scattering potentials, the quantum evolution may result in excitation of the IX internal degrees of freedom, dissociation of the pair, or transmission in small periodic IX wavepackets due to dwelling of one particle in the barrier region. We discuss the occurrence of each process in the full parameter space of the scattering potentials and the relevance of our results for current excitronic technologies. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4905483]

I. INTRODUCTION several micrometers,7,12 while recombination can be controlled by switching off the perpendicular field,13 broadening their Excitonic states are the fundamental below-gap reso- application to an entirely new class of opto-electronic devices. nances in semiconductors heterostructures, where stability In fact, in these devices, data carried by photons can be stored and strength are strongly enhanced by quantum confinement.1 in IXs, elaborated as in charge-based nano-electronics, and Spatially indirect excitons (IXs) are photo-excited correlated finally released back to photons by switching off the perpendic- electron-hole pairs with the two charges localized in different ular electric field at the desired time, all processes being inte- segments of a nano-material. For example, IXs can be easily grated in the same semiconductor segment. Electronics based and selectively excited in semiconductor coupled quantum on excitons is called excitronics. Since gating can be modulated wells (CQWs), where an electric field applied perpendicular to at the GHz frequency, that is, orders of magnitude shorter than the plane of the quantum wells separates positive and negative the IXs intrinsic lifetime, excitronic systems set themselves charges in different layers.2–4 Since optical recombination as a natural bridge between optical data communication and depends on the electron-hole overlap, this extends the intrinsic nano-electronic devices.14–16 lifetime of excitons by orders of magnitude, from nanosec- Several excitronic functions have been demonstrated such onds5 to microseconds,6,7 which allows for the formation of as fast data storage,13,17 acceleration with electrostatic ramps7,18 new types of quantum condensates.6,8–10 and interdigital devices,19 and field effect transistors.12 Most In CQWs with an average inter-well separation d, IXs devices work in a diffusive regime, similar to traditional elec- are two-dimensional quasi-particles in, say, the x,y plane tronics. Therefore, IX is usually described theoretically as a carrying a permanent electric dipole of order −ed( along) the classical gas of interacting dipoles.19,20 However, trapping of growth direction z. Beyond any change in the binding energy single IXs has been recently demonstrated,21 opening to the implied by the particle separation, an electric field Fz shifts realization of quantum excitronic devices, similarly to single 22 the energy of IXs by U = −edFz. Therefore, although elec- electron transistors (SET) in semiconductor quantum dots. trically neutral, IXs couple to any gradient of Fz x,y paral- Clearly, for perspective devices, it is important to deter- lel to the planes.7,11 Such energy landscapes U x(,y )may be mine the behavior of IXs in different regimes, in order to generated by imperfections and defects, but they( can) also be optimize the performance. For example, large dipoles may engineered by properly designed split gates, as in traditional improve acceleration and operation rate, but ensuing smaller nano-electronics, to modulate the perpendicular field in the binding energies may cause dissociation and loss of the stored planes. Within their extended lifetime, IX can be moved over information. IXs are indeed polarizable quasi-particles, with both center-of-mass and internal dynamics. Since in typical systems, applied electrostatic fields in the CQW planes and a)Electronic mail: [email protected] b)Electronic mail: [email protected] the mutual Coulomb interaction of the pair are both in the few c)Electronic mail: [email protected] meV range, the quantum dynamics of IX under the influence

of accelerating potentials is a genuinely two-body process, and (Figure1). Accordingly, we consider the following electron- the internal quantum dynamics cannot be neglected in general. hole Hamiltonian: Although a fundamental problem, the quantum description of ˆ ˆ ˆ ˆ such two-body scattering process has been little discussed in H = T +UC +Uext, (1) 23 the literature. where In this paper we use a unitary propagation scheme to investigate theoretically the exact coherent dynamics of a single pˆ2 pˆ2 Tˆ = e + h , (2) IX wavepacket describing scattering against different types of 2me 2mh electrostatic potential landscapes, namely, steps/downhills and 2 1 e / 4πϵ0 barriers/wells, using a minimal 1D model of an IX (Figure1). In UˆC = − ( ) , (3) ϵ  2 2 this sense, this is a two-particle generalization of the textbook r xê − xˆh + d ( ) description of one particle bouncing against potential steps and Uêxt = Ue xê, t +Uh xˆh, t . (4) barriers. However, since the electron and the hole reside in ( ) ( ) different layers, in typical devices, they are subject to different Here,p ê h , xê h , me h are the 1D linear momentum oper- ( ) ( ) ( ) potentials. Therefore, we have extensively investigated the ator, the coordinate operator, and the effective mass of the ˆ parameter space separately for the two particles, analyzing electron (hole), respectively. UC is the electron-hole Coulomb the time-dependent dynamics during scattering as well as the interaction in a medium of relative dielectric constant ϵ r. ˆ long-time behavior of the scattered wavepacket. In addition to Ue h xê h ,t represents the one-body electrostatic potential ( )( ( ) ) regions where IXs are reflected or transmitted almost as rigid which couples to the electron (hole) coordinatex ê h . Note ( ) objects, we identify experimentally relevant regimes where that in our model, the internal dynamics of the electron-hole scattering is a genuine quantum two-body process. In these pair is described in full, i.e., we do not assume any rigid 24,25 regimes, transmission or reflection of the wavepacket may be exciton approximation, nor the two particles are bound accompanied by excitation of the internal degrees of freedom, by construction: scattering against an external potentials may dissociation of the pair, or transmission in small periodic indeed result in excitation or dissociation of the pair. On the wavepackets due to dwelling of one particle in the barrier other hand, we neglect any lateral extension of the quantum region. states. Energy gaps induced by lateral confinement in a quan- In Sec.II, we set a 1D model of an IX and we briefly tum well or wire (tens of meV) far exceed, and therefore outline the numerical method used for propagation. In Sec. decouple from, the small binding energy of IXs (few meV). III, we analyze the space- and time-dependent dynamics of IX Hence, the main effect of a lateral extension L is to regularize 26 wavepackets scattering against potential steps/downhills and the Coulomb interaction at short distances to 1/L, giving just barriers/wells. In Sec.IV, we discuss the result of scattering a small renormalization of the parameter d. Finally, note that in at asymptotic times in the full parameter space of scattering Ue h xê h ,t , we indicated a possible explicit dependence on ( )( ( ) ) potentials. Our results are discussed in Sec.V in connection time t. Indeed, the numerical methods which we discuss below with experimentally relevant regimes for current technologies. also apply to the case of time-dependent potentials, although Details of the numerical method are given in AppendixA. in this work, we limit ourselves to stationary ones. Below, we shall discuss simulations performed with two paradigmatic classes of electrostatic potentials, namely, poten- II. THEORETICAL APPROACH tial steps/downhills and potential barriers/wells. In Sec.V, we discuss the experimental relevance of these idealized potential A. The Hamiltonian profiles. A smooth potential step in the electron (hole) coordi- To study the dynamics of IXs, we use a 1D model of nate is described as a Fermi-like profile an electron-hole pair propagating under the effect of the two- e h 1 particle mutual Coulomb interaction and external electrostatic Ue h xe h = U0 ( ) − − . (5) ( )( ( )) 1+e xe h b /a potentials coupling to the electron and hole separately. The two ( ( ) ) particles are tight to separated parallel channels at a distance d 1 e h xe h = b is the position of the step, Ue h b = 2U0 ( ). For ( ) e h ( )( ) a particle hitting from the left, U0 ( ) > 0 represents a repulsive e h potential energy step, while U0 ( ) < 0 represents an attractive potential energy downhill. a is a smoothness parameter. Using a smooth potential avoids inaccuracies in numerical work, particularly when numerical derivatives need to be computed and gives a better model of realistic potential modulations.27 For simplicity, we assume common values a and b for electrons and holes. A potential barrier is described by a double Fermi-like profile 1 1 e h − Ue h xe h = U0 ( ) − − + − 1 . ( ) ( )  xe h b1 /a + xe h b2 /a  ( ) 1+e ( ( ) ) 1+e ( ( ) ) FIG. 1. Schematics of the 1D model assumed in the present work. (6)

Here, b1, b2 are the mid-potential positions for the rising e h and descending parts of the barrier, respectively. U0 ( ) > 0 e h represents a potential barrier, while U0 ( ) < 0 represents a potential well. In the absence of any external potential, namely, for a free IX, it is natural to adopt the center of mass (CM) and relative coordinate system m x +m x X = e e h h FIG. 2. Colormap of the potential landscape for an electron-hole pair in the me +mh (7) presence of a barrier potential for the electron and a well for the hole with a = 2 nm, b = 150 nm, b = 190 nm, U e = +3.0 meV, U h = −2.0 meV x = xe − xh 1 2 0 0  as in Eq. (6). (a): xe, xh representation. (b): X, x representation. Also  indicated is the initial( wavepacket,) in arbitrary units,( used) in the simulations since the free Hamiltonian Hˆ0 = Tˆ +UˆC splits into Hˆ0 = HˆCM ˆ  as described in Sec. III. The arbitrary origin is set at the center of the initial + Hr, where IX wavepacket. Pˆ2 Hˆ = (8) CM 2M x = x . On the contrary, in the X,x coordinate frame [Fig. and e h 2(b)], the Coulomb interaction only( ) depends on the relative 2 2 pˆ 1 e / 4πϵ0 coordinate x, while the external potentials Uê and Uˆh couple Hˆ r = − √ ( ) . (9) 2m ϵ r xˆ2 + d2 the X,x coordinates and therefore are represented by the two oblique( ) stripes, the one with negative (positive) slope acting Here, P, p are the CM and relative momenta, respectively, on the electron (hole) alone. Note the difference in the absolute while M ≡ m + m and m ≡ m m /M are the total and the e h e h values of these slopes, ensuing from the very different effective reduced masses. The free IX wave function can be factorized masses. as −iEt/~ Ψ X,x,t = ΦK X φn x e , (10) ( ) ( ) ( ) B. Numerical propagation where the functions Φ X and φ x are eigenfunctions of K n The quantum propagation of the IX is performed on a Hˆ and Hˆ with eigenenergies( ) E ( )and E , respectively. CM r CM,K r,n discrete homogeneous time grid, evolving the state between The CM component is labelled by the CM wavevector K and two consecutive times t and t + ∆ by applying the evolution the relative component by the integer quantum number n. The t operator Uˆ t +∆t,t to the wave function zero of energy is set at the energy gap of the quantum well ( ) material, i.e., the energy for generating the two free particles Ψ x,t +∆t = Uˆ t +∆t,t Ψ x,t . (11) at rest. Therefore, E = ECM,K + Er,n is the total energy of the ( ) ( ) ( ) IX. The evolution operator is applied by using the Split-Step In the presence of an external potential, there is no general Fourier (SSF) method, which is numerically exact for van- 28,29 solution in closed form, since coordinates X,x are coupled ishing ∆t. Although this method is commonly exploited ( ) by Uêxt. On the other hand, had we used the xe,xh coor- for the propagation of a single-particle wavepacket in a multi- ( ) dinates which are separable in Uêxt, they would have been dimensional physical domain, in Eq. (11), we set x = X,x , ( ) coupled by UˆC. In principle, the coherent propagation of a i.e., we interpret the two degrees of freedom of the system as IX wavepacket can be carried out in both coordinate frames. the CM and relative coordinates of the electron-hole couple. However, as we aim at studying scattering events, we find the The SSF method relies on the Suzuki-Trotter expansion30 X,x representation more convenient to analyze the result at to factorize the evolution operator as a product between expo- asymptotic( ) times. Using the X,x set of coordinates turns out nential operators, each of which is diagonal either in direct to be also beneficial from the( )numerical point of view. In fact, or in Fourier space (details are given in AppendixA). For electrons and holes are usually in the same region of space, due this reason, this numerical method relies on a massive use of to Coulomb attraction. As we use a real space representation, Fast Fourier Transformation (FFT), this resulting into a high it is sufficient to describe small regions in r, that is, re ∼ rh, efficiency and a relatively low computational cost in compar- rather than the full xe,xh space. ison with other methods. The use of FFT also implies periodic ( ) Comparing the potential UˆC +Uê +Uˆh in the two different boundary conditions in x and X. However, the simulation coordinate systems may be useful to interpret the results of domain is chosen sufficiently large that the wave function Secs. III andIV. In Fig.2, we show the total potential in the does not reach the boundaries in our analyses. Even if in this case of a narrow barrier for the electron and a well for the work, we have been only concerned about time-independent hole, together with the square modulus of the initial IX wave potentials, the SSF method is also suitable for time-dependent function. The broad negative strip represents the Coulomb ones. potential, while the narrow straight lines are the electron and hole single particle potentials, Uê and Uˆh. In the xe,xh coor- ( ) C. Initial state dinate frame [Fig. 2(a)], Uê and Uˆh are represented by straight stripes parallel to the axes, while the Coulomb interaction cou- The time-dependent analysis starts from the choice of a ples the two coordinates, being maximum along the diagonal realistic initial state. In a photo-generated exciton gas, the

interaction with phonons and other scattering mechanisms typical GaAs parameters, as this is the material of choice allows for exciton relaxation to the ground state φ0 x within a for IX generation and propagation in CQW structures: me ( ) few fs, i.e., on a much shorter timescale than photo-recombina- = 0.067 m0, mh = 0.45 m0, where m0 is the free electron mass, tion lifetime. Therefore, at t = 0, we consider IXs to be in their and ϵ r = 12.9, corresponding to an effective Bohr radius aB 21 ground state φ0 x and we choose = 11.7 nm. All simulations are performed with d = 17 nm. ( ) The eigenvalues of the IX free 1D Hamiltonian Hr obtained Ψ X,x,t = 0 ≡ χ X φ0 x , (12) ( ) ( ) ( ) by straightforward numerical diagonalization are reported in where φn x , already defined in Sec.II, are computed numer- TableI. ically via( finite-di) fference diagonalization, and where χ X is The initial wavepacket has been chosen as described in a minimum uncertainty wavepacket ( ) Sec.IIC with a most probable CM kinetic energy of ECM = 0.5 meV. Typical grid point densities of 2.5 and 0.75 points/nm 1/4 1 X − X 2 have been used for the x and X coordinates, respectively. Sharp χ X = exp − 0 exp iK X , (13) 2 ( 2 ) 0 potential profiles have been used with a = 2 nm. Propagation ( ) 2πσX  4σX  [ ]   has been usually performed up to 40 ps, with a time step ∆t * +   centered in the initial CM position X0 with width σX, and = 20 fs, with x,X ∈ −0.6,0.6 µm × −1.5,1.7 µm. propagating with, an average- CM momentum ( ) [ ] [ ]  2M A. Potential step K ≡ E , 0 2 CM (14) ~ In this subsection, we analyze the scattering against poten- where ECM is the most probable CM kinetic energy of the IX tial steps or downhills, as described by Eq. (5). It turns out with total mass M. that there are four typical phenomenologies, which we discuss below. D. Analysis of the time-dependent dynamics In order to analyze quantitatively the time-dependent dy- 1. Total reflection namics of the IX, we compute the quantum probability h In Fig. 3(a), we show a typical evolution with U0 > ECM andUe = 0.Thiscaseresemblesthetextbookexampleofaparti- 2 0 PΩ t ≡ dxdX Ψ X,x,t . (15) cle reflected by a potential step, with interference fringes form-  ( ) Ω | ( )| ing between incoming and reflected parts of the wavepacket. By choosing the integration domain, Ω, as a suitable After the process is completed, the square modulus of the sub-domain of the simulation region (details are given in wavepacket takes the original Gaussian shape, spread in the Sec. III), we can quantify the transmission, reflection, and CM coordinate according to the free single-particle propaga- dissociation probabilities. tion, with the standard deviation which can be obtained analyti- Furthermore, we analyze excitations of internal degrees of  2 32 callytobe σX t = σX t = 0 1+ t~/ 2MσX t = 0 . This freedom of the IX, which may take place due to the coupling ( ) ( ) e{ [h ( )]} is the general behavior when U0 + U0 & ECM. Although it is between the internal degrees of freedom and the scattering in principle possible to transfer kinetic energy from the CM potential, projecting the wavepacket on the eigenfunctions to the relative motion and excite IX internal levels; for the ˆ φn x of the free Hamiltonian H0. Therefore, at any desired present parameters, this does not happen, since the CM kinetic time( )t, we compute31 energy ECM is substantially smaller than the lowest excitation gap Er,1 − Er,0 = 2.46 meV of the relative coordinate. cn X,t ≡ φn x Ψ X,x,t dx, (16) ( ) Ω ( ) ( ) where Ω is, as in Eq. (15), a suitable sub-domain. This quantity 2. Transmission is further squared to eliminate complex phases and averaged In Fig. 3(b), we show the complementary situation of over the CM coordinate, i.e., an electron undergoing the acceleration by a downhill, while

2 2 the hole does not experience any potential. For this partic- cn X,t ≡ cn X,t dX. (17) | ( )| X  | ( )|

Moreover, we normalize the obtained coefficients, namely, TABLE I. Lowest six out of the eight negative eigenenergies of free IX 2 relative-coordinates Hamiltonian Eq. (9) for the parameters used in the simu- cn X,t c 2 t ≡ | ( )| X . (18) lations (see text). n  2 ⟨| | ⟩( ) m cm X,t X | ( )| n Er, n meV ( ) 0 −4.63 III. SPACE- AND TIME-DEPENDENT DYNAMICS 1 −2.18 2 −1.26 We performed extensive numerical investigations of the 3 −0.80 wavepacket propagation in a broad range of the potentials 4 −0.55 e h intensities, U0 and U0 , considering both the attractive and 5 −0.40 the repulsive case. All simulations have been performed with

FIG. 3. Evolution of an IX wavepacket scattering against a potential step/downhill set at b = 150 nm. The origin of the CM coordinate is set at the center e h e − h of the t = 0 wavepacket. In all cases, only one particle is accelerated: (a) U0 = 0.0 meV, U0 = +2.0 meV. (b) U0 = 4.0 meV, U0 = 0.0 meV. (c) e − h Ω U0 = 5.0 meV, U0 = 0.0 meV. In (b) and (c), labels mark the integration domains used in Eqs. (16) and (15) (see text), limited between dashed lines corresponding to xe = xh = b (slanted lines) and x = ±15aB (vertical lines). A short animation of the continuous time evolution in (a) is included in the supplementary material.33

ular case, almost all the wavepackets are transmitted, as one is the transmission probability. Note that different excited would expect for a single particle scattering. However, here, states are activated at different times and the most advanced the transmission process is more complex. At intermediate part is of the n = 0 character. times, for ≈12 ps, part of the wavepacket is strongly displaced from x = 0, that is, the electron moves far from the hole, while 3. Dissociation the CM remains almost still (due to the larger hole mass, the Similarly to excitation to higher IX states, dissociation CM position corresponds nearly to the hole coordinate). In this into an unbound electron-hole pair is possible if transient, the electron is accelerated, while the hole is not. At a later time, due to Coulomb interaction, the electron drags the hole along. In a classical picture, a wide oscillatory motion of the two particles is activated some picoseconds after the collision. In quantum terms, this corresponds to the excitation of higher quantum levels of the excitonic complex, which can be seen from the complex shape of the transmitted wavepacket in Fig. 3(b). To analyze quantitatively the evolution, in Fig.4, we 2 show the projection coefficients, cn t . The average in the | | ( ) X coordinate is performed over the transmission region A indicated in Fig. 3(b). The scattering starts at t ≈ 3 ps. Higher internal levels are increasingly excited and the scattering process is completed after ≈20 ps. The largest part of the wavepacket (≈40%) is in the first excited (n = 1) state, while another ≈40% is almost equally contributed by states n = 0 and 2 2. All contributions sum up to ≈96% of the wavepacket, which FIG. 4. Evolution of cn t [see Eq. (18)], with Ω ≡ A [see Fig. 3(b)]. ⟨| | ⟩( ) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 034701-6 Grasselli, Bertoni, and Goldoni J. Chem. Phys. 142, 034701 (2015)

e h Er,0 + ECM − U0 +U0 > 0, (19) step, joins the hole at a small kinetic energy, and the whole ( ) ( ) IX is reflected backwards (4), while part of the electron wave that is, an amount of energy exceeding the IX binding energy function is reflected from the step, dragging the hole along, − EB = Er,0 can be transferred to the relative degree of freedom. and the exciton as a whole is transmitted forward (5). However, ≪ In our simulations, since ECM EB, we can observe disso- there is a probability that, after reflection, the electrons do not ciation only with − Ue + Uh . 4.1 meV. In our numerical ( 0 0 ) pull the hole far from the barrier. In this case, the electron per- evolution, dissociation occurs when the wavepacket asymptot- forms one more oscillation, and the process starts over again. ically departs from x = 0 after scattering. In order to quantify Therefore, in such a process, reflection and transmission occur this phenomenon, we compute PΩ≡A t , i.e., we integrate the 3( ) as a periodic pumping of smaller wavepackets. The description probability density in a region far from x = 0 [see Fig. 3(c)], of the process in Fig. 6(a) is analogous with the roles of the and we define the dissociation probability electron and the hole inverted. P ≡ dxdX Ψ X,x,t > t 2, (20) diss  ∞ A3 | ( )| B. Potential barrier where t∞ is a sufficiently large time. We now investigate the scattering from a finite range In Fig. 3(c), a part of the wavepacket is pulled off to the potential, Eq. (6). A barrier/well for each carrier is present right. Here, Pdiss ≈ 30%, as shown in Fig.5. Note that in this in a region b2 − b1 = 40 nm wide. Note that while the width h e particular simulation U0 = 0, while U0 > EB. Therefore, the of the barrier/well has an obvious effect on the scattering electron acquires a high amount of energy from the downhill event, its absolute position is unimportant, as far as the two potential, this exciting the internal dynamics of the IX which parameters b1 and b2 are the same for the two particles. In the eventually dissociates. This is similar (with role of the electron present case, quantum tunneling becomes possible, as in usual h and the hole inverted) to Fig. 3(b), where, however, U0 < EB single-particle scattering. On the other hand, in the asymp- and the IX is excited but does not dissociate. e h totic regions, U0 = U0 = 0. Therefore, it is not possible to excite internal levels or to have free particles, since, as discussed in Sec. III A, the CM kinetic energy which could 4. Dwelling nearby X = b be transferred to the internal degrees of freedom is much As a last interesting phenomenology arising from the two- lower than the excitation gaps of the IX and, a fortiori, its body dynamics, we analyze cases in which part of the exciton binding energy. However, an oscillatory motion or even the wavepacket dwells in the proximity of X = b, i.e., near the step, dissociation of the IX may occur if one of the particle is for a long time interval. This occurs when both the electron and captured by its well, transferring a large energy to the partner the hole experience a strong potential but with opposite sign, a particle. step for the electron and a downhill for the hole or vice versa. In Fig. 7(a), a wavepacket hits a high barrier for electrons Figures 6(a) and 6(b) show the two cases. Dwelling may occur and a weaker well for the hole and tunnels with ≈19% probability. Note that the hole dwells into its potential well in the in region C1 [Fig. 6(a)] or C2 [Fig. 6(b)], depending on the electron/hole potential. A semiclassical interpretation of the initial stages of the process. Dwelling may also lead to periodic process is given in Fig. 6(c) with reference to the case shown emission of the wave function, as for the step/downhill case, in Fig. 6(b). As the IX wavepacket hits scattering potential which is shown in Fig. 7(b) and sketched in Fig. 7(c). When (1), the electron is accelerated along the downhill, while the the IX hits barrier/well potential (1), the electron may be hole does not overcome the repulsive step despite the Coulomb captured by its well, while the hole starts to oscillate around attraction from partner particle (2). Then, the electron, which well position (2). At each oscillation, the wavepacket located is lighter, is pulled backwards by Coulomb interaction (3). in the well region is partially transmitted and partially reflected Part of its wave function is transmitted by the now-repulsive as a IX bound state (3). These oscillations are highlighted by the step-like evolution of the transmission coefficient, shown in Fig.8.

IV. SCATTERING PHASE DIAGRAM The space- and time-dependent evolutions discussed in Sec. III are selected examples in a large set of simulations, to highlight how the scattering process takes place in different regimes. In this section, we summarize the outcome of the scattering events at asymptotic times in the full parameter e h space U0 and U0 . For each simulation, we computed transmission, reflection, and excitation probabilities at t = 40 ps,34 that is, at a sufficiently large time for the scattering process to be completed. These coefficients are used to identify distinct phenomenol- FIG. 5. Evolution of PΩ t [see Eq. (15)], with Ω coinciding with regions ( ) ogies taking place at different values of the potential param- A2, A3, or the central region B indicated in Fig. 3(c). The probability over the e h region A1 is vanishing and it is not shown. eters U0 and U0 . For each class of potentials, either steps/

e h − e − h FIG. 6. As in Fig.3 with (a) U0 = +3.0 meV, U0 = 2.0 meV. (b) U0 = 3.5 meV, U0 = 1.5 meV. (c) Semiclassical interpretation of the scattering process shown in (b). Dashed lines represent equations xe = b and xh = b.

downhills or barriers/wells, we summarize our results in a A. Potential steps/downhills phase diagram as a function of the potential total intensity e h e h 1. Transmission/reflection U0 + U0 and potential asymmetry U0 − U0 between the ( ) e h ( ) two particles, rather than U0 and U0 separately, as the former In Fig. 9(a), the transmission probability is reported as a quantities are those driving the distinct behaviors. A common gray scale colormap. Simulations are performed on a coarse e h e h energy scale for all simulations is set by the kinetic energy grid with steps of 1 meV in U0 + U0 and U0 − U0 , i.e., ( e ) h ( ) ECM = 0.5 meV. In the phase diagrams to be discussed below, in steps of 2 units in the ratios U0 ±U0 /ECM. This choice it is convenient to report results in terms of the dimension- of the simulation space is a good( trade-o)ff between a suffi- e h e h less ratios U0 + U0 /ECM, U0 − U0 /ECM. Note that for a ciently detailed insight and the computational cost. In Fig. rigid exciton,( i.e., an) exciton( with frozen) internal degrees of 9(a), each of the 99 coloured squares corresponds to a specific freedom,24 the scattering outcome (i.e., transmission prob- simulation running for about 8 h (including postprocessing) ability) scales with these ratios, similarly to the textbook on a high-end workstation. When the IX wave function leaves single particle scattering. However, in the present case of a the central region of the domain, i.e., the active region where quantum system with an internal degree of freedom, the latter the external potential may cause reflections or trigger excited couples to the CM kinetics and is affected by the external internal states, the transmission probability is computed by potentials. Therefore, here, the scaling is only approximate: means of Eq. (15). The integration region Ω is indicated as A2 although still a convenient representation, one should keep in in Fig. 3(c) and corresponds to the probability of finding both mind that different behaviors might be expected in a given the electron and the hole beyond the step (xe > b and xh > b) region of the phase diagram if ECM varied. This is particularly and less than 15aB apart (x < 15aB). true for what concerns the phenomenologies arising from As might be expected, transmission is large on the left part the internal dynamics, i.e., dissociation, excitation to higher of the diagram, corresponding to electron and hole downhills, levels, and pausing/pumping nearby the external potential and drops on the right hand side, which correspond to repul- e h region. sive steps for both particles, vanishing at U +U /ECM ≃ 1. ( 0 0 ) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 034701-8 Grasselli, Bertoni, and Goldoni J. Chem. Phys. 142, 034701 (2015)

e h − e − h FIG. 7. Scattering through potentials barriers/wells with b1 = 150 nm, b2 = 190 nm. (a) U0 = 3.5 meV, U0 = 1.5 meV. (b) U0 = 4.0 meV, U0 = 0.0 meV. (c) Sketch of the periodic emission of the IX wavepacket. Short animations of the continuous time evolution in (a) and (b) are included in the supplementary material.33 Note that if the potential is symmetric for the two particles, and holes, so that similar potential intensities, but with the e h i.e., U0 −U0 ≃ 0, the transmission rapidly drops from 1 at role of the particles inverted, have a different effect on the e ( h ) U0 +U0 /ECM ≃ 0 to zero. However, for strongly asymmetric transmission. ( ) e h potentials U0 −U0 /ECM & 5, that is, in the upper and lower parts of the|( diagram,) the| transmission starts to decrease at e h 2. Excitation/dissociation negative values of the total potential U0 + U0 and evolves smoothly to zero. This is because in these regions, one parti- Excitations of the relative motion eigenstates φn x are cle experiences a strongly attractive downhill potential, while e h e h ( ) favored when both U0 + U0 < 0 and U0 − U0 < 0 [lower the other is subject to a weaker but repulsive potential. Note left corner of Fig. 9(a)( ], which) correspond( to a large) downhill also that the diagram is slightly asymmetric in the upper and potential for the electron and a weaker potential step for the lower parts. This is due to the different masses of electrons hole. Clearly, the situation may be reversed, with electron e h and hole potentials inverted, U0 −U0 > 0 (top left corner). However, in this case, the excitation( is) typically weaker, so that regions of excitations and dissociation move to stronger (positive) anisotropies in Fig. 9(a). This asymmetry can be understood from inertial consid- erations: since the hole is one order of magnitude heavier than the electron, a step/downhill for the hole almost coincides with the potential acting on the CM coordinate [a different way to see this is to note that the potential step for the hole is nearly independent from the relative coordinate, compare Figs. 3(a) with 3(b) and 3(c)], and the IX accelerates as a single, rigid particle. On the other hand, when an electron experiences an equivalent downhill potential, a larger part of the external energy can be transferred to the internal relative dynamics. FIG. 8. Transmission coefficient calculated from Eq. (15) with Ω = A [see Dissociation can also take place, when the electron downhill 35 Fig. 7(b)]. is particularly strong.

FIG. 9. Phase diagram for (a) steps/ downhills and (b) barriers/wells simulations. The transmission probability is displayed in grayscale. Simulations are e h performed every 1 meV in U0 + U0 e− h ( ) and U0 U0 . The main phenomenologies( are highlighted) by superimposed letters and solid lines to mark approximate boundaries of different regions. Regions D: dissociation probability greater than about 5%. Regions E: excitation to higher internal levels greater than about 10%. Regions P: periodic transmission/reflection of the wave function due to dwelling. The positive (negative) slope dashed line indicates h e U0 ( ) = 0.

B. Potential barriers/wells this analysis applies to 1D channels (e.g., electrostatically defined coupled quantum wires), it allows to capture themain Contrary to the step/downhill case, with a barrier/well phenomenologies to be expected also in planar CQWs. Next, potential, it is not possible to have excitation of internal levels we discuss the connection to realistic systems and perspective nor dissociation in the transmitted part of the wavepacket. As experiments. discussed in Sec. III B, however, for Ue +Uh /E ≪ 0 and 0 0 CM For the prototypical potential profiles considered here (but Ue h ≫ E , dissociation in the barrier( region) takes place if 0 ( ) CM in a large range of parameters) IX scattering times turn out to one| particle| (the electron or the hole, depending whether Ue 0 be in the order of tens of ps. This is smaller than the lifetime or Uh is negative and intense) is captured into its well, while 0 of IXs in CQWs and also their coherence time in quantum the other inertially continues its motion. wells,36 and supports the possibility to engineer single-IX Due to quantum tunneling, the transmission probability phase coherent excitronic devices. is finite also for Ue + Uh /E ≥ 1. Note that tunneling is 0 0 CM Having used typical material parameters and potential substantially anisotropic( between) the upper part (Ue > 0 and 0 landscapes for GaAs-based structures, we expect that the Uh < 0) and the lower part (Ue < 0 and Uh > 0) of the phase 0 0 0 complex phenomenologies found in our simulations could take diagram. Again, this is clearly due to the difference between place in excitronic devices and should be taken into account electron and hole effective masses, as for large anisotropies, in device engineering. Moreover, the complex time-dependent IX does not tunnel as a rigid complex of mass M, but rather IX dynamics exposed by our results could be directly probed, through a complex two-body process. On the other hand, as for example, by collecting time-resolved micro-photolumines- one may expect from one-body tunneling, resonant transmis- cence maps7,12 during the scattering process, exploiting the sion may take place. This is shown in Fig. 9(b) by the oscil- fact that optical recombination of IXs can be triggered at a latory behavior of the transmission coefficient, particularly in 17,37 desired time by switching off the vertical field Fz. Lumi- the upper part of the phase diagram. nescence maps could be calculated38–40 from Ψ X,x = 0 2, Finally, note that, at difference with the steps/downhills as provided, e.g., by our simulative method. Although| ( we did)| case where dissociation and excitation take place more easily not attempt an explicit calculation of optical signatures to be for negative anisotropies (lower left part of the phase diagram) directly compared to experiments, since our model is strictly than for positive anisotropies (upper left part), here, the oppo- 1D, for purpose of illustration we provide in the supplementary site is true, since these phenomena happen by the trapping material33 the real-space and K-space evolution of Ψ X,x = 0 . mechanism, which is much more favorable for holes. Note that in this respect single-IX devices could offer( a unique) possibility to probe the quantum two-body scattering dynamics by optical means. In realistic 2D devices, scattering V. CONCLUSIONS times could be larger than those found in our simulations, due We have discussed the unitary space- and time-dependent to the larger phase space. Time-resolution can thus be less quantum dynamics of a single IX in a CQW system scattering demanding. against simple potential profiles. Our analysis suggests, in Potential profiles as those employed in the present study particular, that exciton scattering in typical devices is a genuine are usually induced by electrostatic gates. On the one hand, two-body process, substantially different and more complex typical top gates induce an electrostatic potential in the under- than the single-particle scattering of a rigid exciton. Our results lying 2D system with opposite sign for the two particles. Due are based on a minimal 1D model. Although strictly speaking, to layer separation d, however, the potential generated by a top

gate is different in magnitude for the two layers. For example, is to exploit the evolution operator factorization, Eq. (A3), in for a device of total thickness 1 µm with 1 V applied to top order to act on the state function in the specific representation gates the resulting voltage difference between the two CQWs in which the operator is diagonal. At each time step, the SSF 10 nm apart is of the order of 10 mV. Therefore, a common algorithm performs the following operations: situation in excitronic devices is with large potential anisot- 1. Multiplication of the wave function in coordinates repre- ropies, corresponding to the upper and lower sectors between sentation by the rightmost potential term, for half time step the bisecting lines in Fig.9. ( ∆ ) ∆ On the other hand, it is also possible to gate separately − i Uˆ x,t+ t t Ψ x,t −→ Ψ′ x = e ~ 2 2 Ψ x,t . (A4) the two layers of the heterostructure and engineer potential ( ) ( ) ( ) landscapes of the same sign and similar intensities for the 2. Computation of the Fourier transformation of the state two particles.41 Potential profiles of the same sign for the two Ψ′ x particles with few meV anisotropies correspond to left and ( ) right regions between the bisecting lines in Figs.9. Scatter- Ψ˜ ′ p = dxe−ip·xΨ′ x . (A5)  ing potentials of the same sign may also arise from mono- ( ) ( ) layer fluctuations of the QW thickness which induce modu- 3. Multiplication of the wave function in momentum space by lations of the lateral confinement energy in the meV range the kinetic energy term for a full time step, ∆t

i.e., a steps/downhills potential profiles along the planes of the p2 i  i ′ ′′ − i ∆t ′ QWs. Ψ˜ p −→ Ψ˜ p = e ~ 2mi Ψ˜ p . (A6) ( ) ( ) ( ) 4. Calculation of the inverse Fourier transform of the wave ACKNOWLEDGMENTS function 1 We thank M. Rontani for a critical reading of the manu- Ψ′′ x = dpe+ip·xΨ˜ ′′ p . (A7) script and L. Sorba for useful discussions. We acknowledge ( ) 2π 2  ( ) ( ) partial financial support from EU-FP7 Initial Training Net- 5. Multiplication of the wave function by the leftmost poten- work INDEX and University of Modena and Reggio Emilia, tial term for another half time step Grant “Nano- and emerging materials and systems for sustain- ( ∆ ) ∆ − i Uˆ x,t+ t t able technologies.” ′′ ~ 2 2 ′′ Ψ x −→ Ψ x,t +∆t = e Ψ x . (A8) ( ) ( ) ( ) The SSF is a unitary method which preserves the norm of 28 APPENDIX A: THE TWO-PARTICLE SPLIT-STEP the wave function. We underline that in Eq. (A3), operators METHOD Tˆ and Uˆ are represented in their respective diagonal forms using the appropriate basis. Therefore, even though a spatial In order to study the time-dependent scattering dynamics discretization of the domain (as described in Sec. III) is used of IXs, we use a numerically exact solution of the Time in order to represent the wavepackets and the potential U, no Dependent Schrödinger Equation (TDSE) with the Hamilto- finite ffdi erence approach is needed to represent the kinetic nian given by Eq. (1) based on the Split Step Fourier method. operator, which instead is represented in the reciprocal grid The time propagation of a generic quantum state Ψ x,t , ( ) implied by the discrete Fourier transform algorithm. between two close time instants, t and t +∆t, can be expressed The SSF method is commonly used to propagate a wave- by means of the evolution operator, Uˆ t +∆t,t , as ( ) packet representing one particle in an n-dimensional space, e.g., with x ≡ x,y for n = 2. However, since it is an exact Ψ x,t +∆t = Uˆ t +∆t,t Ψ x,t , (A1) ( ) ( ) ( ) solution of the( Schrödinger) equation, there is no limitation where as to the interpretation of the set of coordinates x. Here, we  ∆  i t+ t implement the method by propagating the two-body wave Uˆ t +∆ ,t = T exp − Tˆ +Uˆ τ dτ t ~  function in the x ≡ X,x coordinates. Note that the momentum ( ) t ( ) ( )  ( ) vector p ≡ P,p has also components corresponding to the CM i ( ∆t ) ( ) ≈ T exp − Tˆ∆t +Uˆ t + ∆t . (A2) and relative momenta, respectively. ~ 2 Unlike the isotropic mass one-body case, the two coordi- Here, T exp is the so-called time ordered exponential nates are governed by different terms in the Hamiltonian, with operator. In the{} last line, the “rectangle integration method” very different effective masses, resulting in different frequency has been used for integrating the potential energy. The last line ranges for P and p. While typical energies for the relative in Eq. (A2) is exact for time independent potentials. motion are in the few meV range, the typical kinetic energy By using the Suzuki-Trotter formula30 in order to expand for the CM is in the fraction of an meV. It is this decoupling the exponential of the sum of operators in Eq. (A2) as a product which makes particularly convenient to work in the CM and of exponential operators, we obtain relative coordinates.

i ˆ ∆t ∆t i ˆ i ˆ ∆t ∆t ˆ − ~ U t+ − ~ T ∆t − ~ U t+ 3 U t +∆t,t = e ( 2 ) 2 e e ( 2 ) 2 + O ∆t . (A3) ( ) ( ) APPENDIX B: WAVEPACKET IN K SPACE We now exploit the fact that the operator Tˆ is diagonal in Fourier space, while the operator Uˆ is diagonal in direct space. In Fig. 10, we show the K-space representation of Ψ X,x The basic idea behind the Split Step Fourier (SSF) method = 0 for the three cases (and the same selected times) presented( ) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 034701-11 Grasselli, Bertoni, and Goldoni J. Chem. Phys. 142, 034701 (2015)

FIG. 10. K-space representation of the wave function at four different times for the same IX scatterings shown in Figs. 3(a), 7(a), and 7(b), in columns (a), (b), and (c), respectively.

in Figs. 3(a), 7(a), and 7(b). This quantity is directly related to 3K. Sivalertporn, L. Mouchliadis, A. L. Ivanov, R. Philp, and E. A. Muljarov, the luminescence intensity38–40 which is ∝ Ψ X,x = 0 2. The Phys. Rev. B 85, 045207 (2012). 4 highlighted region around the origin satisfies| ( )| A. A. High, A. T. Hammack, J. R. Leonard, S. Yang, L. V. Butov, T. Ostatnický, M. Vladimirova, A. V. Kavokin, T. C. H. Liew, K. L. Campman, n and A. C. Gossard, Phys. Rev. Lett. 110, 246403 (2013). K < E , (B1) c~ g ap 5M. Colocci, M. Gurioli, A. Vinattieri, F. Fermi, C. Deparis, J. Massies, and G. Neu, Europhys. Lett. 12, 417 (1990). 2 √ which represents the K-space radiative cone. Here, n = ϵ r is 6L. V. Butov, J. Phys.: Condens. Matter 16, R1577 (2004). the refractive index of GaAs, c the speed of light, ~ the reduced 7A. Gärtner, A. W. Holleitner, J. P. Kotthaus, and D. Schuh, Appl. Phys. Lett. Planck constant and Eg ap = 1.424 eV the GaAs energy gap. 89, 052108 (2006). The videos movieK_fig3a.mpg, movieK_fig7a.mpg, 8L. Keldysh and A. Kozlov, J. Exp. Theor. Phys. 27, 521 (1968). andmovieK_fig7b.mpg of the supplementary material33 show 9A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler, L. V. Butov, A. the time evolution of Ψ X,0 2 (top panel) and its K-space V. Kavokin, K. L. Campman, and A. C. Gossard, Nature 483, 584 (2012). | ( )| 10M. Alloing, M. Beian, M. Lewenstein, D. Fuster, Y. González, L. González, transform (bottom panel) for the three cases of supplementary R. Combescot, M. Combescot, and F. Dubin, Europhys. Lett. 107, 10012 Fig. 1. The K components that comply with Eq. (1) are also (2014). highlighted. The evolution shows that such optically active 11M. Hagn, A. Zrenner, G. Böhm, and G. Weimann, Appl. Phys. Lett. 67, 232 components are substantially modified during the scattering of (1995). 12 the IX. A. A. High, E. E. Novitskaya, L. V. Butov, M. Hanson, and A. C. Gossard, Science 321, 229 (2008). movieXx_fig3a.mpg movieXx_ Moreover, the videos , 13A. G. Winbow, A. T. Hammack, L. V. Butov, and A. C. Gossard, Nano Lett. fig7a.mpg and movieXx_fig7b.mpg of the supplementary 7, 1349 (2007). material33 show the continuous-time evolution of Ψ X,x 2 14A. A. High, A. T. Hammack, L. V. Butov, M. Hanson, and A. C. Gossard, for the three scattering events also shown in Figs. 3(a)| (, 7(a))|, Opt. Lett. 32, 2466 (2007). and 7(b). 15Y. Y. Kuznetsova, M. Remeika, A. A. High, A. T. Hammack, L. V. Butov, M. Hanson, and A. C. Gossard, Opt. Lett. 35, 1587 (2010). 16P. Andreakou, S. V. Poltavtsev, J. R. Leonard, E. V. Calman, M. Remeika, Y. 1G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures Y. Kuznetsova, L. V. Butov, J. Wilkes, M. Hanson, and A. C. Gossard, Appl. (Les Editons de Physique, Paris, 1988), Chap. IV, and references therein. Phys. Lett. 104, 091101 (2014). 2L. Butov, C. Lai, A. Ivanov, A. Gossard, and D. Chemla, Nature 417, 47 17A. G. Winbow, L. V. Butov, and A. C. Gossard, J. Appl. Phys. 104, 063515 (2002). (2008).

18J. R. Leonard, M. Remeika, M. K. Chu, Y. Y. Kuznetsova, A. A. High, L. 29A. Castro, M. A. L. Marques, and A. Rubio, J. Chem. Phys. 121, 3425 V. Butov, J. Wilkes, M. Hanson, and A. C. Gossard, Appl. Phys. Lett. 100, (2004). 231106 (2012). 30M. Suzuki, Proc. Jpn. Acad., Ser. B 69, 161 (1993). 19 31 A. G. Winbow, J. R. Leonard, M. Remeika, Y. Y. Kuznetsova, A. A. High, The φn x can be chosen to be real valued functions. A. T. Hammack, L. V. Butov, J. Wilkes, A. A. Guenther, A. L. Ivanov, M. 32G. I. Márk,( ) Eur. J. Phys. 18, 247 (1997). Hanson, and A. C. Gossard, Phys. Rev. Lett. 106, 196806 (2011). 33See supplementary material athttp: //dx.doi.org/10.1063/1.4905483 for an 20J. Wilkes, E. A. Muljarov, and A. L. Ivanov, Phys. Rev. Lett. 109, 187402 animation of real-space and K-space wavefunction evolution. (2012). 34For selected simulations, when the IX dissociates, we stopped the 21G. J. Schinner, J. Repp, E. Schubert, A. K. Rai, D. Reuter, A. D. Wieck, A. O. simulation at shorter times, i.e., when one of the particles was hitting Govorov, A. W. Holleitner, and J. P. Kotthaus, Phys. Rev. Lett. 110, 127403 the boundary of the simulation range. (2013). 35In practice, in the simulations IX dissociation is signalled by excitation 22M. Kastner, Ann. Phys. 9, 885 (2000), and references therein. of higher and higher internal levels, since an increasing number of bound 23A. Arkhipov and Y. Lozovik, Phys. Lett. A 319, 217 (2003). states is be necessary to describe the free part of the wavepacket as time 24R. Zimmermann, F. Große, and E. Runge, Pure Appl. Chem. 69, 1179 proceeds. (1997). 36A. Enderlin, M. Ravaro, V. Voliotis, R. Grousson, and X.-L. Wang, Phys. 25U. Hohenester, G. Goldoni, and E. Molinari, Appl. Phys. Lett. 84, 3963 Rev. B 80, 085301 (2009). (2004). 37Y. Lozovik and I. Ovchinnikov, Solid State Commun. 118, 251 26A. Esser, R. Zimmermann, and E. Runge, Phys. Status Solidi B 227, 317 (2001). (2001). 38J. Feldmann, G. Peter, E. O. Göbel, P. Dawson, K. Moore, C. Foxon, and R. 27This is not important in our simulation due to the diagonal representation J. Elliott, Phys. Rev. Lett. 59, 2337 (1987). of the kinetic operator, but it may be critical for a posteriori calculation of 39L. C. Andreani, F. Tassone, and F. Bassani, Solid State Commun. 77, 641 average quantities, like the kinetic energy. (1991). 28C. Leforestier, R. Bisseling, C. Cerjan, M. Feit, R. Friesner, A. Guldberg, A. 40L. Butov, A. Gossard, and D. Chemla, Nature 418, 751 (2002). Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, 41J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. 58, 1497 and R. Kosloff, J. Comput. Phys. 94, 59 (1991). (1991).